Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials

We consider the Hill operator

Equiconvergence of spectral decompositions of Hill operators

We study in various functional spaces the equiconvergence of spectral decompositions of the Hill operator L = ?d ²/dx ² + v(x), x ∈ L ¹([0, π], with H _per ^?1 -potential and the free operator L ⁰ = ?d ²/dx ², subject to periodic, antiperiodic or Dirichlet boundary conditions. In particular, we prove that $\left\| {S_N - S_N^0 :L^a \to L^b } \right\| \to 0if1 < a \leqslant b < \infty ,1/a - 1/b < 1/2,$ , where S _N and S _N ⁰ are the N-th partial sums of the spectral decompositions of L and L ⁰. Moreover, if v ∈ H ^?α with 1/2 < α < 1 and $\frac{1} {a} = \frac{3} {2} - \alpha $ , then we obtain the uniform equiconvergence ‖S _N ?S _N ⁰ : L ^a → L ^∞‖ → 0 as N → ∞.     

The divergence of spectral decompositions connected with homogeneous elliptic operators

In this work an asymptotic formula is obtained for the means of the Riesz spectral function of a homogeneous elliptic operator of arbitrary order m≥2 with constant coefficients. Conditions are obtained under which localization (and convergence) of the means of the Riesz spectral decompositions of functions from the Hölder class is absent.     

Rational integrability of trigonometric polynomial potentials on the flat torus

We consider a lattice ? ? ? ⁿ and a trigonometric potential V with frequencies k ∈ ?. We then prove a strong rational integrability condition on V, using the support of its Fourier transform. We then use this condition to prove that a real trigonometric polynomial potential is rationally integrable if and only if it separates up to rotation of the coordinates. Removing the real condition, we also make a classification of rationally integrable potentials in dimensions 2 and 3 and recover several integrable cases. After a complex change of variables, these potentials become real and correspond to generalized Toda integrable potentials. Moreover, along the proof, some of them with high-degree first integrals are explicitly integrated.     

On decompositions of trigonometric polynomials

Let R _t [θ] be the ring generated over R by cosθ and sinθ, and R _t (θ) be its quotient field. In this paper we study the ways in which an element p of R _t [θ] can be decomposed into a composition of functions of the form p = R ? q, where R ∈ R(x) and q ∈ R _t (θ). In particular, we describe all possible solutions of the functional equation R ₁ ? q ₁ = R ₂ ? q ₂, where R ₁,R ₂ ∈ R[x] and q ₁, q ₂ ∈ R _t [θ].     

On scalar extensions and spectral decompositions of complex symmetric operators

In this paper we prove that a complex symmetric operator with property (δ) is subscalar. As a corollary, we get that such operators with rich spectra have nontrivial invariant subspaces. We also provide various relations for spectral decomposition properties between complex symmetric operators and their adjoints.     

Equiconvergence of spectral decompositions of 1D Dirac operators with regular boundary conditions

One dimensional Dirac operators $L_{bc}\left(v\right)y=i\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill ?1\hfill \end{array}\right)\frac{dy}{dx}+v\left(x\right)y\text{,}y=\left(\begin{array}{c}\hfill y_1\hfill \\ \hfill y_2\hfill \end{array}\right)\text{,}x\in [0,\pi ]\text{,}$ considered with $L^2$-potentials $v\left(x\right)=\left(\begin{array}{cc}\hfill 0\hfill & \hfill P\left(x\right)\hfill \\ \hfill Q\left(x\right)\hfill & \hfill 0\hfill \end{array}\right)$ and subject to regular boundary conditions ($bc$), have discrete spectrum. For strictly regular $bc$, the spectrum of the free operator $L_{bc}\left(0\right)$ is simple while the spectrum of $L_{bc}\left(v\right)$ is eventually simple, and the corresponding normalized root function systems are Riesz bases. For expansions of functions of bounded variation about these Riesz bases, we prove the uniform equiconvergence property and point-wise convergence on the closed interval $[0,\pi ]$. Analogous results are obtained for regular but not strictly regular $bc$.     

A criterion for Hill operators to be spectral operators of scalar type

We derive necessary and sufficient conditions for a Hill operator (i.e., a one-dimensional periodic Schrö dinger operator) H = ?d ² /dx ² + V to be a spectral operator of scalar type. The conditions show the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential V. In the course of our analysis, we also establish a functional model for periodic Schrödinger operators that are spectral operators of scalar type and develop the corresponding eigenfunction expansion.The problem of deciding which Hill operators are spectral operators of scalar type appears to have been open for about 40 years.     

Loaded differential operators: Convergence of spectral expansions

We study the convergence rate of biorthogonal series expansions of functions in systems of root functions of a broad class of loaded even-order differential operators defined on a finite interval. These expansions are compared with the Fourier trigonometric series expansions of the same functions in an integral metric on any interior compact set of the main interval or on the entire interval. We obtain estimates for the equiconvergence rate of these expansions.     

Numerical solution of inverse spectral problems for Sturm-Liouville operators with discontinuous potentials

We consider Sturm-Liouville differential operators on a finite interval with discontinuous potentials having one jump. As the main result we obtain a procedure of recovering the location of the discontinuity and the height of the jump. Using our result, we apply a generalized Rundell-Sacks algorithm of Rafler and Böckmann for a more effective reconstruction of the potential and present some numerical examples.     

Localization conditions for spectral decompositions related to elliptic operators from the classA r

We study sufficient localization conditions for spectral decompositions of distributions from negative Sobolev classes with compact supports. Sufficient localization conditions are obtained for spectral decompositions corresponding to elliptic differential operators with constant coefficients.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 421–427, March, 1996.The author wishes to express gratitude to Prof. Sh. A. Alimov for his attention to this work.     

Hyponormality of trigonometric Toeplitz operators

In this paper we establish a tractable and explicit criterion for the hyponormality of arbitrary trigonometric Toeplitz operators, i.e., Toeplitz operators with trigonometric polynomial symbols . Our criterion involves the zeros of an analytic polynomial induced by the Fourier coefficients of . Moreover the rank of the selfcommutator of is computed from the number of zeros of in the open unit disk and in counting multiplicity.

     

Asymptotic expansion of norm associated with conjugate trigonometric polynomial

LetT _n (x) be trigonometric polynomial of degreen, and be conjugation ofT _n (x). In this paper we obtain the complete asymptotic expansion for

On oscillatory integral Hilbert transformation with trigonometric polynomial phase

Let $ \mathcal{P}_n $ denote the set of algebraic polynomials of degree n with the real coefficients. Stein and Wpainger [1] proved that $$ \mathop {\sup }\limits_{p( \cdot ) \in \mathcal{P}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{ip(x)} }} {x}dx} } \right| \leqslant C_n , $$ where C _n depends only on n. Later A. Carbery, S. Wainger and J. Wright (according to a communication obtained from I. R. Parissis), and Parissis [3] obtained the following sharp order estimate $$ \mathop {\sup }\limits_{p( \cdot ) \in \mathcal{P}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{ip(x)} }} {x}dx} } \right| \sim \ln n. $$ . Now let $ \mathcal{T}_n $ denote the set of trigonometric polynomials $$ t(x) = \frac{{a_0 }} {2} + \sum\limits_{k = 1}^n {(a_k coskx + b_k sinkx)} $$ with real coefficients a _k , b _k . The main result of the paper is that $$ \mathop {\sup }\limits_{t( \cdot ) \in \mathcal{T}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{it(x)} }} {x}dx} } \right| \leqslant C_n , $$ with an effective bound on C _n . Besides, an analog of a lemma, due to I. M. Vinogradov, is established, concerning the estimate of the measure of the set, where a polynomial is small, via the coefficients of the polynomial.     

Local maxima of a random trigonometric polynomial

In this paper we obtain a formula for the expected number of maxima of a normal process (t) which occur below a levelu. The main condition assumed in the derivation is that (t) and its first and second derivative have, with probability one, continuous one-dimensional distributions. The expected number of such maxima of a trigonometric polynomial with random coefficients follow from this result. It is shown that, wich probability one, all the maxima of this type of polynomial occur below a levelu if , and a sizeable number of maxima exist below zero level.